Average Error: 12.1 → 1.2
Time: 21.6s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{\mathsf{fma}\left(\frac{t}{z}, -\frac{y}{2}, z\right)}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y}{\mathsf{fma}\left(\frac{t}{z}, -\frac{y}{2}, z\right)}
double f(double x, double y, double z, double t) {
        double r315982 = x;
        double r315983 = y;
        double r315984 = 2.0;
        double r315985 = r315983 * r315984;
        double r315986 = z;
        double r315987 = r315985 * r315986;
        double r315988 = r315986 * r315984;
        double r315989 = r315988 * r315986;
        double r315990 = t;
        double r315991 = r315983 * r315990;
        double r315992 = r315989 - r315991;
        double r315993 = r315987 / r315992;
        double r315994 = r315982 - r315993;
        return r315994;
}
↓
double f(double x, double y, double z, double t) {
        double r315995 = x;
        double r315996 = y;
        double r315997 = t;
        double r315998 = z;
        double r315999 = r315997 / r315998;
        double r316000 = 2.0;
        double r316001 = r315996 / r316000;
        double r316002 = -r316001;
        double r316003 = fma(r315999, r316002, r315998);
        double r316004 = r315996 / r316003;
        double r316005 = r315995 - r316004;
        return r316005;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original12.1
Target0.1
Herbie1.2
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 12.1

    \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]

  2. Simplified1.2

    \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(\frac{t}{z}, -\frac{y}{2}, z\right)}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity1.2

    \[\leadsto \color{blue}{1 \cdot \left(x - \frac{y}{\mathsf{fma}\left(\frac{t}{z}, -\frac{y}{2}, z\right)}\right)}\]

  5. Final simplification1.2

    \[\leadsto x - \frac{y}{\mathsf{fma}\left(\frac{t}{z}, -\frac{y}{2}, z\right)}\]

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :herbie-target
  (- x (/ 1 (- (/ z y) (/ (/ t 2) z))))

  (- x (/ (* (* y 2) z) (- (* (* z 2) z) (* y t)))))